G. S. Canright

The crystal problem for polytypes

Recent work on discrete classical problems in one-dimensional statistical mechanics has shown that, given certain elementary symmetries, such problems may not have a periodic (crystalline) ground state, even in the absence of fine tuning of the couplings. Here these results are applied to several families of well known polytypic materials. The families studied are those represented by the compounds SiC, CdI(2) and GaSe, and also the micas and kaolins. For all families but SiC, it is found that there is a finite probability for the ground state to be degenerate and disordered.

∊-Machine spectral reconstruction theory: a direct method for inferring planar disorder and structure from X-ray diffraction studies

In previous publications [Varn et al. (2002). Phys. Rev. B, 66, 174110; Varn et al. (2007). Acta Cryst. B63, 169-182] we introduced and applied a new technique for discovering and describing planar disorder in close-packed structures directly from their diffraction patterns. Here, we provide the theoretical development behind those results, adapting computational mechanics to describe one-dimensional structure in materials. We show that the resulting statistical model of the stacking structure - called the ε-machine - allows the calculation of measures of memory, structural complexity and configurational entropy. The methods developed here can be adapted to a wide range of experimental systems in which power spectra data are available.

DEFECT-UNBINDING TRANSITIONS AND INHERENT STRUCTURES IN TWO DIMENSIONS

We present a large-scale (36thinsp000-particle) computational study of the {open_quotes}inherent structures{close_quotes} (IS) associated with equilibrium, two-dimensional, one-component Lennard-Jones systems. Our results provide strong support both for the inherent-structures theory of classical fluids, and for the Kosterlitz-Thouless-Halperin-Nelson-Young theory of two-stage melting in two dimensions. This support comes from the observation of {ital three} qualitatively distinct {open_quotes}phases{close_quotes} of inherent structures: a crystal, a {open_quotes}hexatic glass,{close_quotes} and a {open_quotes}liquid glass.{close_quotes} We also directly observe, in the IS, analogs of the two defect-unbinding transitions (respectively, of dislocations and disclinations) believed to mediate the two equilibrium phase transitions. Each transition shows up in the inherent structures, although the free disclinations in the {open_quotes}liquid glass{close_quotes} are embedded in a percolating network of grain boundaries. The bond-orientational correlation functions of the inherent structures show the same progressive loss of order as do the three equilibrium phases: long-range {r_arrow} quasi-long-range {r_arrow} short-range. {copyright} {ital 1998} {ital The American Physical Society}

Inferring planar disorder in close-packed structures via epsilon-machine spectral reconstruction theory: structure and intrinsic computation in zinc sulfide.

We apply epsilon-machine spectral reconstruction theory to analyze structure and disorder in four previously published zinc sulfide diffraction spectra and contrast the results with the most common alternative theory, the fault model. In each case we find that the reconstructed epsilon-machine provides a more comprehensive and detailed understanding of the stacking structure, often detecting stacking structures not previously found. Using the epsilon-machines reconstructed for each spectrum, we calculate a number of physical parameters - such as configurational energies, configurational entropies and hexagonality - and several quantities - including statistical complexity and excess entropy - that describe the intrinsic computational properties of the stacking structures.

Exclusion statistics for fractional quantum Hall states on a sphere

We discuss exclusion statistics parameters for quasiholes and quasielectrons excited above the fractional quantum Hall states near � = p/(2np + 1). We derive the diagonal statistics parameters from the (“unprojected”) composite fermion (CF) picture. We propose values for the off-diagonal (mutual) statistics parameters as a simple modification of those obtained from the unprojected CF picture, by analyzing finite system numerical spectra in the spherical geometry.

Inferring planar disorder in close-packed structures viaɛ-machine spectral reconstruction theory: examples from simulated diffraction patterns

A previous paper detailed a novel algorithm, ε-machine spectral reconstruction theory (εMSR), that infers pattern and disorder in planar-faulted, close-packed structures directly from X-ray diffraction patterns [Varn et al. (2013). Acta Cryst. A69, 197-206]. Here εMSR is applied to simulated diffraction patterns from four close-packed crystals. It is found that, for stacking structures with a memory length of three or less, εMSR reproduces the statistics of the stacking structure; the result being in the form of a directed graph called an ε-machine. For stacking structures with a memory length larger than three, εMSR returns a model that captures many important features of the original stacking structure. These include multiple stacking faults and multiple crystal structures. Further, it is found that εMSR is able to discover stacking structure in even highly disordered crystals. In order to address issues concerning the long-range order observed in many classes of layered materials, several length parameters are defined, calculable from the ε-machine, and their relevance is discussed.

On the spectral problem for anyons

We consider the spectral problem resulting from the Schrödinger equation for a quantum system ofn≧2 indistinguishable, spinless, hard-core particles on a domain in two dimensional Euclidian space. For particles obeying fractional statistics, and interacting via a repulsive hard core potential, we provide a rigorous framework for analysing the spectral problem with its multi-valued wave functions.

Nature of ergodicity breaking in ising spin glasses as revealed by correlation function spectral properties

In this Letter we address the nature of broken ergodicity in the low temperature phase of Ising spin glasses by examining spectral properties of spin correlation functions C(ij) identical with. We argue that more than one extensive [i.e., O(N)] eigenvalue in this matrix signals replica symmetry breaking. Monte Carlo simulations of the infinite-range Ising spin-glass model, above and below the Almeida-Thouless line, support this conclusion. Exchange Monte Carlo simulations for the short-range model in four dimensions find a single extensive eigenvalue and a large subdominant eigenvalue consistent with droplet model expectations.

Disordered ground states for classical discrete-state problems in one dimension

It is known that one-dimensional lattice problems with a discrete, finite set of states per site “generically” have periodic ground states (GSs). We consider slightly less generic cases, in which the Hamiltonian is constrained by either spin (S) or spatial (I) inversion symmetry (or both). We show that such constraints give rise to the possibility ofdisordered GSs over a finite fraction of the coupling-parameter space—that is, without invoking any nongeneric “fine tuning” of coupling constants, beyond that arising from symmetry. We find that such disordered GSs can arise for many values of the number of statesk at each site and the ranger of the interaction. The Ising (k=2) case is the least prone to disorder:I symmetry allows for disordered GSs (without fine tuning) only forr≥5, whileS symmetry “never” gives rise to disordered GSs.

Extensive eigenvalues in spin-spin correlations: A tool for counting pure states in Ising spin glasses

We study the nature of the broken ergodicity in the low temperature phase of Ising spin glass systems, using as a diagnostic tool the spectrum of eigenvalues of the spin-spin correlation function. We show that multiple extensive eigenvalues of the correlation matrix C ij [^SiS j& occur if and only if there is replica symmetry breaking. We support our arguments with Exchange Monte Carlo results for the infinite-range problem. Here we find multiple extensive eigenvalues in the replica symmetry breaking~RSB! phase for N*200, but only a single extensive eigenvalue for phases with long-range order but no RSB. Numerical results for the short-range model in four spatial dimensions, for N<1296, are consistent with the presence of a single extensive eigenvalue, with the subdominant eigenvalue behaving in agreement with expectations derived from the droplet model. Because of the small system sizes we cannot exclude the possibility of replica symmetry breaking with finite size corrections in this regime.